Questions about the properties of functions of the form $\sum_{n=0}^{\infty}a_n x^n$, where the $a_n$ are real or complex numbers, and $x$ is real or complex (or more generally an element of a Banach algebra).

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1answer
22 views

If the limit of power series exists, it converges.

Let $f(x) = \sum a_n x^n$ converges on $(-R, R)$. Does $\sum a_n R^n$ converge if $\lim _{x \to R-} f(x)$ exists?
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1answer
31 views

Find the radius of convergence about the origin

I need to find the radius of convergence about the origin for this function $$ G(z) = \left(\frac{1 - \sqrt{1-4abz^2}}{2}\right) $$ where, $$ \\ a+b = 1, \\ 0 < b < \frac 12 $$ I'm finding it ...
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0answers
43 views

First develop the function $\sqrt{x}$ in a series of powers of $(x-1)$ and then use it to approximate $\sqrt{0.9999999995}$ to ten decimal places. [on hold]

First develop the function $\sqrt{x}$ in a series of powers of $(x-1)$ and then use it to approximate $\sqrt{0.9999999995}$ to ten decimal places. I'm stuck on how to do this problem. Any solutions ...
2
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1answer
68 views

Power series expansion of an Operator.

I've been reading a paper called "Separation of variables for the quantum $Sl(2,R)$ spin chain" in which the author at one point does a power series expansion I do not understand. The problem is this ...
2
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0answers
16 views

Fractional Euler sums?

As we know, the classical linear double Euler sums is defined by $${S_{p,q}} = \sum\limits_{n = 1}^\infty {\frac{{{\zeta _n}\left( p \right)}}{{{n^q}}}} \;$$ where $p, q\ (q \ge 2)$ are positive ...
2
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2answers
37 views

What function is this? $\sum_{k=0}^\infty \frac{2^{2k}z^{2k-1}}{(2k)!}$ [on hold]

I am interested to know what function represents the following series: $$\sum_{k=0}^\infty \frac{2^{2k}z^{2k-1}}{(2k)!}$$
0
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1answer
389 views

Taylor Series Theorem

So I see the argument presented in taylor series, that $$\sum c_n (x-a)^n = \sum \frac{f^{(n)}(a)}{n!} (x-a)^n$$ or $c_n = f^{(n)}(a)/n!$ if $x=a$ the question is, since the above only holds when ...
28
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3answers
793 views

On calculating $\int_0^1\ln(1-x^2)\;{\mathrm dx}$ — where is the mistake?

I've seen the integral $\displaystyle \int_0^1\ln(1-x^2)\;{dx}$ on a thread in this forum and I tried to calculate it by using power series. I wrote the integral as a sum then again as an integral. ...
4
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2answers
61 views

If $\sum_{m,n}a_{mn}x^m(1-x)^n\equiv 0$, can we conclude $a_{mn}=0$?

Assume $\{a_{mn}\}$ are some real numbers between -1 and 1. If we know $$\sum_{m,n}a_{mn}x^m(1-x)^n\equiv0\quad\forall x\in(0,1),$$ can we conclude that $a_{mn}=0$ for all $m,n\geq 0?$ Thanks.
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6answers
59 views

What is $1+2+4+8+16+…+2^n$? [duplicate]

What is the result of: summation from one, two, four, eight until $n$ power of two? Thank you!
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0answers
33 views

integral of complex function, power series

let $\mu$ be a finite borel measure on $[0,+\infty)$ and let $f$ be defined by $$f(z)=\int_{[0,+\infty)}\frac{d\mu(t)}{t-z},\quad z \in \mathbb{C} \setminus [0,+\infty)\,.$$ *show that the integral ...
0
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1answer
31 views

Series expansion for innocent looking function

$$f(z) = \frac{1}{z^2}$$ is given, where $f(z)$ is complex valued function. How can one find series expansion at $ z=i$ with using geometric series approach? It seems simple but first tries gives ...
2
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0answers
188 views

Algorithm for reversion of power series?

Given a function $f(x)$ of the form: $$f(x) = x/(a_0x^0+a_1x^1+a_2x^2+a_3x^3+a_4x^4+a_5x^5+...a_nx^n)$$ Let $A$ be an arbitrary (any) infinite lower triangular matrix with ones in the diagonal: $$A ...
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3answers
42 views

Evaluating the ratio $ {{a_{n+1}}\over{a_n}}$ in calculating the radius of convergence for a power series

In calculating the radius of convergence for the power series $$ \sum_{n=1}^\infty {{(2n)!}\over(n!)^2}\ x^n $$ By the ratio test, we let $$ a_n = \lvert {{(2n)!}\over(n!)^2}\ x^n \rvert \quad\quad ...
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2answers
41 views

About complex power series

I have a really big doubt. I'm trying to find all the values of $z$ for which the next power series converges: $$\sum_{n=0}^{\infty} \frac{z^{3n}}{8^{n}(1-in)} $$ Using the root test I have that ...
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3answers
38 views

Existence of an analytic function under some given conditions

Which of the followings is(/are) correct? There exists an entire function $f:\mathbb C \to \mathbb C$ which takes only real values & is such that $f(0)=0$ & $f(1)=1$. There exists an ...
0
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1answer
496 views

Y-Axis units on FFT graph

A 50Hz sinusoid wave with a voltage range of +/-20V is sampled at 512Hz for 1 second. No bias or phase shift are present. The signal is run through an FFT. The result is one spike at 50Hz on the ...
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0answers
47 views

Positivity of an alternating series.

Greetings esteemed mathematicians. I've managed to prove that the following series \begin{equation} f_{\lambda}(\omega)= ...
54
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2answers
5k views

Fractal behavior along the boundary of convergence?

The complex power series $$\sum_{n=1}^{\infty}\frac{z^{n^2}}{n^2}$$ has radius $1$ (Ratio Test) and is absolutely convergent along $|z|=1$. Recalling something that my calculus professor (Ray Mayer, ...
3
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1answer
74 views

Finding a power series representation for $\left(\frac{x}{2-x}\right)^3$

Find a power series representation for $\displaystyle\left(\frac{x}{2-x}\right)^3$ My approach is in finding something similar to $\displaystyle\left(\frac{x}{2-x}\right)^3$ to which I can easily ...
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0answers
33 views

Crossed homomorphisms between power series groups

Consider the group $\mathbb{C}[[z]]_1$ of the power series of the form $a_1 z + a_2 z^2 + \cdots$, with $a_1\neq 0$, under the operation of composition, and $\mathbb{C}[[z]]$ as a ...
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3answers
185 views

How to calculate the limit of this sum with different methods? [duplicate]

It's a basic question , but what are the common methods to calculate limits like this one: $$\sum_{k=1}^\infty \frac{3k}{7^{k-1}}$$
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1answer
659 views

Equation about generating functions and subfactorial

Suppose $G_n(w)$ is a formal power series (really a probability generating function, see the following explanation) of variable $w$, try to solve out $G_n(w)$ for all $n\ge0$ from the ...
14
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3answers
173 views

Show that $\frac{x}{3!}-\frac{x^3}{5!}+\frac{x^5}{7!}-\cdots\leq \frac{1}{\pi}$.

My problem is to show that $$\frac{x}{3!}-\frac{x^3}{5!}+\frac{x^5}{7!}-\cdots\leq \frac{1}{\pi}$$ for all $x\in\Bbb R$. I was thinking of first finding the max and then show that its less ...
0
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1answer
59 views

How to calculate this sum?

Let $x_1,\cdots,x_k$ be numbers between 0 and 1. Then is it possible to get explicit expression for the following sum:$$\sum_{n_1,\cdots,n_k\geq 1} x_1^{n_1}\times C_{n_1+n_2}^{n_2}\times ...
0
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1answer
78 views

Taking Limits of Sets

I know this sounds like a ridiculous idea- but it's the only one I can think of for this radius of convergence problem for a power series involving sine. I want to let $P:= \{k:|sin(k)| \geq \delta ...
1
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1answer
32 views

Power series of z/sin(z)?

So I need to compute the coefficient of the $z^4$ in the power series of $\frac{z}{\sin z}$. I tried differentiating the function and obtaining coefficients like in Taylor's expansions but had a ...
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4answers
127 views

Prove the series has positive integer coefficients

How can I show that the Maclaurin series for $$ \mu(x) = (x^4+12x^3+14x^2-12x+1)^{-1/4} \\ = 1+3\,x+19\,{x}^{2}+147\,{x}^{3}+1251\,{x}^{4}+11193\,{x}^{5}+103279\, ...
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1answer
48 views

Multiplicative inverse of the power series $e^x - c$ for $c \neq 1$.

We know that the power series $f(x)= e^x -c \in \mathbb C[[x]]$ for $c \neq 1$, has a multiplicative inverse, since it's constant coefficient is non-zero. I was wondering whether the inverse is known ...
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3answers
51 views

Why radius of power series is defined as $\lim_{n \to \infty} (a_n)^{1/n} = 1/R$

I am reading definition of radius of convergence of power series $a_nx^{n}$ as $\limsup_{n \to \infty} (|a_n|)^{1/n} = 1/R$. I cannot understand it intutively, it makes no sense to me. Can anyone ...
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1answer
75 views

Can you define the radius of convergence of a power series by an upper bound on the sequence of coefficients?

Let $P(z) = \sum_{n = 0}^\infty c_n z^n$ be a complex power series. Consider the follwing subsets of $\mathbb{R}$ $$ \begin{align} A_1 &:= \{r \geq 0 \,:\, (c_n r^n)_{n \in \mathbb{N}_0} \text{ ...
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0answers
22 views

Simple example of approximating a nonlinear system with Volterra series

I'm trying to understand Volterra series as a means of modelling/approximating nonlinear input-output relations. I'm having trouble to understand the abstract definitions of kernels/functionals and ...
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1answer
28 views

Finding a power series representation for $\frac{1}{(2-x)^2}$ in powers of $x$

Problem: Determine a power series representation for the function \begin{align*} \frac{1}{(2-x)^2} \end{align*} in powers of $x$. On what interval is the representation valid? Attempt: We have ...
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2answers
55 views

Are there many different power series representation for a given function?

So I have to find the power series representation for $f(x) = \ln (3-x)$. I attempted the following: $$\ln(3-x) = \int {- \frac{1}{3-x} dx}$$ $$ = - \int { \frac{1}{1-(x-2)} dx}$$ $$ = - \int ...
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2answers
191 views

Power series expression for $\exp(-\Delta)$

I know it should be true, but for some reason I can't get the calculations to work out in order to show that if $f$ is smooth and compactly supported, the power series $\sum_{j=0}^\infty ...
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0answers
56 views

We have $ f(x) = \sum_{n \geq 1} \frac{(x-1)^n}{n}$ prove that $f(x) = -\ln(2-x)$.

I am having problems with the following exercise, I have solved the first two parts of the exercise but I am unsure about the last part. I have the following power series $$f(x) = \sum_{n \geq 1} ...
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3answers
45 views

Differentiate the following power series $\sum_{n \geq 1} \dfrac{(2x-2)^n}{n2^n+1}$

I am having issues with the differentiation of the following power series $$ \large f(x) = \sum_{n \geq 1} \dfrac{(2x-2)^n}{n2^n+1}$$ I get the following result $$ \large f'(x) = \sum_{n \geq 1} ...
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2answers
51 views

Find $\sum_{n=1}^{\infty}a_nx^n$ given $a_0=3, \ 3na_n+3(n-1)a_{n-1}=2a_{n-1}$

Given $\ a_0=3$, $\,3na_n+3(n-1)a_{n-1}=2a_{n-1}$, find $\ f = \sum_{n=1}^{\infty}a_nx^n$. I have proved that when $\ \left\lvert x \right\rvert<1$, this exponential series function is convergent. ...
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0answers
42 views

Summation of an infinite series

For $0<\theta\leq 1$ and $A, B \geq 1$, we wish to find summation (or upper bound) of the following infinite series: $$ \frac{(\theta)^2 }{A^{(\theta^{\frac{1}{2}})}B^{(\theta^{\frac{1}{2}})}} ...
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5answers
157 views

How to show that $e^{-x}$ tends to $0$ when $x\to \infty$ if $e^{-x}$ is defined as the power series.

With only the formal definition of $$f(x) = \exp(-x)= \sum \frac{(-x)^n}{n!}$$ how can we show that $$\lim_{x\to \infty} f(x)=0?$$ I am looking for a proof that would not use the identity ...
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2answers
20 views

Generalized Matrix Series

I have to sum a matrix series of the form: $$ \sum_{s=0}^\infty M^s B R^s $$ Is it possible to obtain a closed form formula as in the usual geometric series? Thanks.
0
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1answer
32 views

coefficients of a power series

I have the function $$f(x)=\frac{2x}{10+x}$$ and I am asked to find its power series representation which I found to be $$\sum_{n=0}^{\infty} (-1)^{n} *\frac{2x^{n+1}}{10^{n+1}}$$ and I found the ...
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1answer
34 views

Finding series representation of $\frac{1}{P(D)}$ through ordinary division

I am studying ODEs from ordinary differential equations by Tenenbaum and Pollard. The book in its fifth chapter explains inverse operators for finding the particular solution of a constant coefficient ...
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2answers
219 views

Sum of 1.5-powers of natural numbers

I recently have met the following approximate equation: $$\sum_{k=1}^n k^{1.5}\approx\frac{n^{2.5}+(n+1)^{2.5}}{5}.$$ It's a rather accurate approximation (for $n=40$ the absolute error is $\approx ...
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3answers
50 views

Being careful with terms of infinite sums

$ cos(x): = \sum_{k=0}^\infty \frac{(-1)^nx^{2n}}{2n!}$ $=1- \frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\frac{x^8}{8!}...$ I would like to show that for $x \in [0,2]$ $cos(x) \leq 1- ...
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1answer
40 views

An upper-bound problem of sum of positive numbers

I came across the following problem of inequality. If $ \ \ \sum_{i=1}^{n}x_i^3\leq S$ then find the value of $K$ such that $\sum_{i=1}^{n}x_i\leq K$. It is given that $x_i>0,\forall i\in ...
0
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1answer
31 views

Solve for power series satisfying certian relations among each other

Write $y_i = f_i(z) = \sum_{k=1}^{\infty} c_k^{(i)} z^k$ for $i=1,\ldots, i$ for four (formal) power series. The following relations are given between them \begin{align*} y_1 & = y_2 + y_3 \\ ...
0
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1answer
39 views

Are these two power series equal?

Let $f(x)=\sum_{n=0}^\infty a_n x^n$ and $g(x)=\sum_{n=0}^\infty b_n x^n$ where $a_n,b_n\in[0,1]$ for all $n\geq 0$. Hence we know these two power series are convergent on $(-1,1)$. Now assume there ...
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0answers
105 views

How did Euler give a sum to the divergent series $…x^{-3}+x^{-2}+x^{-1}+1+x^1+x^2+x^3.. = 0$?

In Prof Norman Wildberger's A Socratic look at the logical weaknesses of modern pure mathematics (which just made available on youtube), he mentioned a discovery by Euler (30:55) that: ...
1
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2answers
62 views

series function

We know that there are some series that can be written in short, for example: $$ \sum_{n=0}^\infty x^n=\frac{1}{1-x},\qquad |x|<1 $$ Is there similar function for $$ \sum_{n=1}^N x^{1/n} $$ or $$ ...